17-2 Traveling Sound Waves#

Prompts

Write the displacement \(s(x,t)\) for a sinusoidal sound wave traveling in the +\(x\) direction. What is \(s_m\)? How does an air element move?
Write the pressure variation \(\Delta p(x,t)\). How does \(\Delta p\) relate to compression vs expansion? What is \(\Delta p_m\)?
How are \(s(x,t)\) and \(\Delta p(x,t)\) related in phase? When the displacement is maximum, what is the pressure variation? When the element passes through equilibrium?
Derive or state the relationship \(\Delta p_m = \rho v \omega s_m\). What does it tell you about loud sounds?

Lecture Notes#

Overview#

A sinusoidal sound wave produces displacement \(s(x,t)\) and pressure variation \(\Delta p(x,t)\) that vary sinusoidally in space and time.
Displacement \(s\): longitudinal position of an air element relative to equilibrium. Pressure variation \(\Delta p\): deviation from atmospheric pressure.
\(s\) and \(\Delta p\) are 90° out of phase—when displacement is maximum, pressure is at equilibrium; when the element passes through equilibrium, pressure is at its extreme.

Displacement#

For a sound wave traveling in the \(+x\) direction:

(121)#\[ s(x,t) = s_m \cos(kx - \omega t) \]

\(s_m\): displacement amplitude—maximum displacement of an air element from equilibrium.
\(k = 2\pi/\lambda\), \(\omega = 2\pi f\); same definitions as for transverse waves (section 16-1).
Each air element oscillates left and right (along \(x\)) in SHM as the wave passes.

Pressure variation#

The pressure at position \(x\) varies from the equilibrium (atmospheric) pressure:

(122)#\[ \Delta p(x,t) = \Delta p_m \sin(kx - \omega t) \]

\(\Delta p_m\): pressure amplitude—maximum increase or decrease in pressure.
\(\Delta p > 0\): compression (density increased).
\(\Delta p < 0\): expansion (rarefaction; density decreased).

Phase relationship and amplitude link#

\(s\) and \(\Delta p\) are 90° out of phase. When \(s\) is maximum (element at extreme displacement), \(\Delta p = 0\)—the element is momentarily at rest and the pressure has returned to equilibrium. When \(s = 0\) (element passing through equilibrium, moving fastest), \(\Delta p\) is at its maximum magnitude—compression or rarefaction is greatest there.

The pressure amplitude and displacement amplitude are related by

(123)#\[ \Delta p_m = \rho v \omega s_m \]

Louder sound → larger \(s_m\) and \(\Delta p_m\). For typical loud sounds, \(s_m\) is still very small (e.g., ~10 µm at the threshold of pain).

Poll: Element at equilibrium

When an air element is moving rightward through its equilibrium position (\(s = 0\)), is the pressure in the element at equilibrium, just beginning to increase, or just beginning to decrease?

(A) At equilibrium
(B) Just beginning to increase
(C) Just beginning to decrease

Summary#

Displacement: \(s = s_m \cos(kx - \omega t)\); air elements oscillate longitudinally.
Pressure variation: \(\Delta p = \Delta p_m \sin(kx - \omega t)\); \(\Delta p > 0\) = compression.
Phase: \(s\) and \(\Delta p\) are 90° out of phase.
\(\Delta p_m = \rho v \omega s_m\)—links pressure and displacement amplitudes.